I have relatively little interest in the BBT, but I would certainly not agree that even if black holes don't exist that such would challenge the Kalam Cosmological Argument (which is the one I assume you are referring to). And I'll further note that I am not much of a fan of the KCA
anyway. All the argument needs in support of its second premise (that the universe came into existence) is evidence in favor the an absolute beginning, which is
not synonymous with beginning in a singularity. For even if singularities don't exit, it doesn't follow therefore that the universe always existed.
With that said, this is why I think we need to be careful how we present the KCA. You can either use it for what I think it really is, which is a scientific argument, or else you can attempt to use it as a demonstration. Frankly, I don't think it works as the latter, and I've explained why in some detail
elsewhere. If, though, you want to use it as a scientific argument, you have to be ready to concede immediatly that future science may challenge the current theories. That does NOT mean that we cannot hold the current theories to be true and therefore draw reasonable inferences from them. It DOES mean that such theories, and thus our inferences, are always open to doubt and correction. BUT that means that they are open precisely to the degree that the theory is questionable. In other words, if someone wants to challenge your belief in God by attacking the KCA, they have to show that there is good evidence that the universe did NOT have a beginning. It is not enough to say, "Well, the theory might be wrong, so you can't prove it!" That fails as a counter argument, because what you can (and should) say is, "Of course the theory might be wrong, but we have no reason to think that it is. In fact, all of the evidence as we currently have it suggests X, and therefore, I am justified in holding X to be true unless and until contrary evidence is discovered. You, however, are not justified in saying that X is false unless and until you can provide evidence that it is false. Therefore, do you have any scientific evidence that the theory is false?"
Now, someday, someone might present just such an argument, and at that point, your argument would be refuted. But if the universe really DID begin in a singularity, then all such evidence will turn out to be wrong on inspection. If, though, the universe did NOT begin in a singularity, you do not want to base your faith in God on something that turns out not to be true, so here, in fact, you can thank the atheist for you doing you a service. In the meantime, I would remind you that showing blackholes do not exist is not sufficient to overturn the standard model of the BBT. Therefore, you are still warranted, scientifically speaking, in holding it to be true and drawing from it appropriate inferences (i.e., that the universe came into existence); and by extension, the atheist is UNWARRANTED, scientifically speaking, in rejecting it and thereby failing to draw appropriate inference (i.e., that the universe came into existence). And therefore, the atheist who rejects the argument is doing so on unwarranted, blind faith. And as they so very well know, you cannot argue with blind faith, so I would tell you not to bother with those who argue from such a stance (as most do, in my experience).
edit:
what Rick said